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Integral question

• To: mathgroup at smc.vnet.net
• Subject: [mg73675] Integral question
• From: Tulga Ersal <tersal at umich.edu>
• Date: Sat, 24 Feb 2007 02:20:22 -0500 (EST)

Dear Mathematica users,

Let's consider the integral

Integrate[1/Sqrt[v^2 - c*s^2], s]

where v and c are positive reals. When I calculate the integral by hand, I get

ArcSin[(Sqrt[c]*s)/v]/Sqrt[c]

However, if I evaluate the integral in Mathematica, I get

(I*Log[(-2*I)*Sqrt[c]*s + 2*Sqrt[-(c*s^2) + v^2]])/Sqrt[c]

which not only does not look like what I have found by hand, but 
also, for v=4, c=2, s=1, for example, gives

0.255525 + 1.47039 i

as opposed to just 0.255525.

If you evaluate the integral in Mathematica with the said values (v=4, c=2)

Integrate[1/Sqrt[4^2 - 2*s^2], s]

you get the answer

ArcSin[s/(2*Sqrt[2])]/Sqrt[2]

which agrees with what I have found by hand.

My question is: How can I get what I found by hand using Mathematica 
without having to assign values to v and c before evaluating the 
integral? I tried adding the assumptions v>0, c>0, but it didn't help.

I'd appreciate your help.

Thanks,
Tulga